Explain quantitatively how an object can follow circular motion whilst on a ramp with no friction in the radial direction.

The important condition for circular motion is that there must be a constant force acting perpendicular to the motion of the object. In this case there will be a contact force acting on the object from the ramp.If the speed of the object is great enough then the contact force will provide both the centripetal force and counteract the weight of the object.The object is then in equilibrium in the radial direction and will not require friction to keep it from going up the ramp.